How do i start a new stroke from an existing anchor point using curvature tool?

Daniel,

With Smart Guides being your friends, what happens if you start by Clicking and holding outside the snapping distance form the Anchor Point, then press the Spacebar and ClickDrag to snap, then ClickDrag as desired?

The curvature tool will always activate an existing path when you click on it.

The next thing is: you can of course start a path in that same place, but you cannot connect it to an existing path. Y-connections are not allowed in bézier paths.